High viscosity limit for the multi-dimensional compressible Navier-Stokes equations

TL;DR

This paper studies the behavior of solutions to the multi-dimensional compressible Navier-Stokes equations in the high viscosity limit, showing convergence to a well-posed transport equation with damping, including cases with variable viscosity.

Contribution

It establishes the high viscosity limit for compressible Navier-Stokes equations with both constant and variable viscosity coefficients, including the nonlocal damping effects.

Findings

Density converges to a solution of a transport equation with nonlinear damping.

Global well-posedness of the limit equation is proved for large data.

Results hold for both constant and variable viscosity coefficients.

Abstract

We investigate the high viscosity limit (also called inertial limit) of the barotropic compressible Navier-Stokes equations supplemented with initial data which are perturbations of a stable constant solution. In the case of constant viscosity coefficients, we establish that, after diffusive rescaling, the density tends to satisfy a transport equation with nonlinear damping which is globally well-posed, even for large data. Similar results are proved for variable viscosity coefficients. In this latter case, the damping term in the limit equation of the density is nonlocal.